for Viscous-Flow Problems

Initial and Boundary Conditions for Viscous-Flow Problems

Zhaoyuan Wang and Fred Stern

IIHR—Hydroscience & Engineering C. Maxwell Stanley Hydraulics Laboratory

The University of Iowa

ME6260 (058:260) Viscous Flow

http://user.engineering.uiowa.edu/~me_260/Viscous_flow_main.htm

Feb. 1, 2021

2

Why are ICs and BCs needed?

• Solutions to ODEs are usually not unique (integration constants exist), which is also a problem for PDE’s.

• PDE’s are usually specified through a set of ICs and BCs.

• A BC expresses the behavior of a function on the

boundary of the domain. An IC specifies the value of the function in time direction, at time t = 0.0.

• The GDEs to be discussed next constitute an IBVP for a

system of 2nd order nonlinear PDE, which require IC and

BC for their solutions, depending on physical problem and

appropriate approximations.

Initial Conditions

• Initial conditions (ICS, steady/unsteady flows)

o ICs should not affect final results and only affect convergence path, i.e. number of

iterations (steady) or time steps (unsteady) need to reach converged solutions.

o More reasonable guess can speed up the convergence

o For complicated unsteady flow problems, CFD codes are usually run in the steady

mode for a few iterations for getting a better

initial conditions

4

Boundary Conditions

Types of BCs: can be defined/categorized mathematically, physically, and numerically.

• Mathematical definitions

• For flow variables

o Kinematic BCs: motion without regard for the cause o Dynamic BCs: the causes of motion

Name Form

Dirichlet 𝜙 = 𝑓

Neumann 𝜕𝜙

𝜕𝑛 = 𝑓 Robin

𝐶₀𝜙 + 𝐶₁𝜕𝜙

𝜕𝑛 = 𝑓

Mixed 𝜙 = 𝑓 , 𝐶₀𝜙 + 𝐶₁^𝜕𝜙

𝜕𝑛 = 𝑓

Cauchy 𝜙 = 𝑓 and 𝐶₀ ^𝜕𝜙

𝜕𝑛 = 𝑔

Boundary Conditions

• Physical domain boundaries:

o Solid Surface

 Fixed, moving wall, Deforming wall, FSI

 Permeable Interface, Porous Surface o Free Surface, Wave Boundary

o Two-Phase Interface Internal Jump conditions o Inlet/exit/outer

o Fairfield/Open

5

-52 44

0 x/H

z/H _Inlet

Non-slip Wall Slip Wall

Jump Conditions Outlet

6

Boundary Conditions

• For grid and numerical treatment:

o Symmetric BC o Periodic BC

o Numerical beach, absorbing BC

o Multiblock/Overset overlapping grid BC o Convection BC

o Pole BC (singularity)

o Global mass conservation enforcing BC

Overset and patched multiblock grids for airfoil.

Symmetric BC Pole BC

Examples of Boundary conditions

1. Solid Surface

 Fixed, moving wall

 Permeable interface, porous surface

 Deforming wall, FSI

2. Single phase flows: Free surface BCs 3. Multiphase flows: Two-phase interface

jump conditions 4. Inlet/exit/outer

Free surface flow: droplet

Two-phase interfacial flow: bubble Pipe flow with no-slip (A) and slip (B) boundary conditions. (Berg et al., 2021).

8

Examples of Boundary conditions

1. Solid Surface

No-slip BCs: No-slip BC widely used for most macroscopic flows without loss of accuracy

• ℓ = mean free path of a moving molecular

particle << fluid motion; therefore, macroscopic view is “no slip” condition, i.e. no relative motion or temperature difference between liquid and solid.

• Exception for gas and contact line problem

liquid solid

V  V T

_liquid

 T

_solid

Smooth wall:

Specular reflection

Conservation of tangential momentum

u_w=0=fluid velocity at wall

Rough wall:

Diffuse reflection. Lack of reflected tangential momentum balanced by u_w

Examples of Boundary conditions

Slip-wall BCs:

w

w

dy

l du

u 

10

Examples of Boundary conditions

Contact line problem:

Contact angles for a droplet

• No-slip BCs used for most macroscopic flows without loss of accuracy but pose a problem in viscous flows at contact lines.

• For small scale flows, slip BC with a finite slip length is usually used: 𝑢₀ = 𝑏 ^𝜕𝑢

𝜕𝑦 . The contact line movement is also dependent to the contact angle, but the mechanism is not fully understood.

• For large scale flows with high Reynolds numbers, very small grid spacing is usually used near the wall in order to resolve the boundary layer.

• In CFDShip-Iowa, a blanking distance (𝑏) is used for the interface functions, which is chosen based on the 𝑦⁺. The recommended value is 𝑦⁺ >30 (outside of the turbulence buffer region ), and 𝑦⁺ = 100 is usually used for most simulations of ship flows according to the numerical experiments.

Wave blanking in CFDShip-Iowa

Examples of Boundary conditions

Permeable interface, porous surface:

Suction or Injection 𝑢 = 0, 𝑛𝑜 𝑠𝑙𝑖𝑝

𝑣 = 𝑣

_𝑠

or 𝑣 = 𝑣

_𝑖

, 𝑓𝑙𝑜𝑤 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡ℎ𝑒 𝑤𝑎𝑙𝑙

𝑇

_{𝑓𝑙𝑢𝑖𝑑}

= 𝑇

_{𝑤𝑎𝑙𝑙}

, no temperature jump

𝑞

_𝑤

= ℎ𝑇

_𝑦

|

_𝑤

= 𝜌

_𝑖

𝑣

_𝑖

𝑐

_𝑝

(𝑇

_𝑤

− 𝑇

_𝑖

), energy at the

wall

12

Examples of Boundary conditions

Fluid Structure Interaction (FSI) BCs:

Kinematic continuity between the fluid and the structure is ensured by the non-slip wall condition:

𝑢_𝑤 = ^𝜕𝑥

𝜕𝑡, 𝑣_𝑤 =^𝜕𝑦

𝜕𝑡 , 𝑤_𝑤 = ^𝜕𝑧

𝜕𝑡

where 𝒖_𝑤 = {𝑢_𝑤, 𝑣_𝑤, 𝑤_𝑤}is the velocity of the fluid particle and 𝒙 = {𝑥, 𝑦, 𝑧}are the coordinates of the solid wall.

The continuity of the momentum is ensured by the continuity of the stress across the fluid-structure interface:

𝜏_𝑖𝑗∙ 𝑛_𝑖|_𝑤 = 𝜏_𝑖𝑗 ∙ 𝑛_𝑖|_𝑠 The energy conservation, considering the first law of thermodynamics

𝛿𝑄 − 𝛿𝑊 = 𝑑𝐸 The energy equation for adiabatic CV,

−^𝛿𝑊

𝑑𝑡 =^𝑑𝐸

𝑑𝑡 = ^𝜕

𝜕𝑡׮_{𝑉 𝑡} 𝑒𝜌 𝑑𝑉 + ׭_{𝑆 𝑡} 𝑒𝜌 𝒖 ∙ ො𝑛 𝑑𝑆

𝑒 = 𝑘_𝑒 + 𝑝_𝑒𝜀 + 𝑝_𝑒𝑔. 𝑘_𝑒 , 𝑝_𝑒𝜀 and 𝑝_𝑒𝑔 are the kinetic and elastic and gravitational potential energies.

The work rate flows are exchanged between the water CV and the structure CV.

ሶ𝑊 = ሶ𝑊_{𝑠ℎ𝑎𝑓𝑡}

𝑝𝑢𝑚𝑝/𝑡𝑢𝑟𝑏𝑖𝑛𝑒

+ 𝑊ดሶ_𝑝

𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒

+ 𝑊ดሶ_𝜈

𝑣𝑖𝑠𝑐𝑜𝑢𝑠

ሶ𝑊_𝑝 and ሶ𝑊_𝑣 are the pressure and viscous work rates done by the CV.

Within the fluid CV, 𝑊ሶ_{𝑠ℎ𝑎𝑓𝑡} = 0. Within the structure CV, 𝑊ሶ_{𝑠ℎ𝑎𝑓𝑡} is the work rate done by the plate’s mount on the system and is hereafter named 𝑊ሶ_𝑀.

The pressure and viscous work done on solid = pressure and viscous work done on fluid and vice versa, which are equivalent to separate dE/dt for solid which includes 𝑒 = 𝑘_𝑒 + 𝑝_𝑒𝜀 + 𝑝_𝑒𝑔 and fluid which only includes 𝑒 = 𝑘_𝑒 + 𝑝_𝑒𝑔. In most cases net outflux of energy from the CV is zero.

Thus for the fluid ^𝑑𝐸

𝑑𝑡 = ^𝜕

𝜕𝑡׮_𝑉

𝑤𝜌_𝑤 𝑔𝑧 + ^{𝒖𝑤 2}

2 𝑑𝑉_𝑤 And for the solid ^𝑑𝐸

𝑑𝑡 = ^𝜕

𝜕𝑡׮_{𝑉 𝑡} 𝑒𝜌 𝑑𝑉

Examples of Boundary conditions

Single flow free surface BCs:

•

Free surface problems since interface is unknown and part of the solution, but effect gas on liquid idealized.

•

Assume the upper fluid (air) is an “atmosphere” that merely exerts pressure on the lower fluid (water), with shear and heat conduction negligible.

•

Kinematic FSBC: free surface is stream surface

•

Dynamic FSBC: stress continuous across free surface (similar for mass and heat flux)

Approximations:

𝑝 ≈ 𝑝_𝑎 = 0, neglect air viscosity and surface tension 𝜉_𝑥~𝜉_𝑥~0, small slope

𝑤_𝑥~𝑤_𝑦~𝑤_𝑧 = 0, ^𝜕𝑢

𝜕𝑧 = ^𝜕𝑣

𝜕𝑧 = 0 small normal velocity gradient

14

Examples of Boundary conditions

Two-phase interface jump conditions:

The velocity fields in fluids 1 and 2 are continuous across the interface if there is no phase change and mass transfer across the interface,

𝐮_𝟏 = 𝐮_𝟐 (1)

where 𝐮 is the velocity vector. The interface velocity 𝑉_𝐼 is the normal velocity and is the same on both sides of the interface:

𝑉_𝐼 = 𝐮_𝟏 ∙ 𝐧 = 𝐮_𝟐 ∙ 𝐧 (kinematic condition) (2) where n is the unit normal vector.

The continuity of the tangential velocities is analogous to the no-slip boundary condition on a wall,

𝐮_𝟏 − 𝐮_𝟏 ∙ 𝐧 𝐧 = 𝐮_𝟐 − 𝐮_𝟐 ∙ 𝐧 𝐧 (continuity of the tangential velocity) (3)

n Fluid 1

Fluid 2

σ u₁

u₂ T₂ T₁

Examples of Boundary conditions

Stress conditions:

The stress tensor is defined in terms of the local fluid pressure and velocity field as

𝐓 = −𝑝𝐈 + 𝛕 = −𝑝𝐈 + 𝜇[∇𝐮 + ∇𝐮 ^𝑇] (4)

where𝐈 is the unit tensor, 𝛕 is viscous stress tensor, 𝑝 is pressure, and 𝜇 is the dynamic viscosity. The stress vector, the force (per unit area) exerted by the fluid on the interface, is defined as,

𝐭 𝐧 = 𝐧 ∙ 𝐓 (5)

Note that the tress vector in the above equation generally includes both the normal and tangential stress components.

The exact interface stress condition is given in the stress balance equation below:

𝐧 ∙ 𝐓_𝟏 − 𝐧 ∙ 𝐓_𝟐 = σ𝐧 ∇ ∙ 𝐧 − ∇σ (6)

where ∇σ is tangential stress associated with gradients of the surface tension. The divergence of the unit normal is related to the mean curvature:

𝛁 ∙ 𝐧 = 𝜅 (7)

The stress jump condition can be rewritten as

𝐧 ∙ (𝐓_𝟏 − 𝐓_𝟐) = σ𝜅𝐧 − ∇σ (8)

16

Examples of Boundary conditions

Note that both normal and tangential stresses must be balanced at the interface. The condition can be written separately as the “normal stress balance” and “tangential stress balance”.

Normal stress balance

Projection of Eq. (8) along the unit normal n obtains,

𝐧 ∙ 𝐓_𝟏 − 𝐓_𝟐 ∙ 𝐧 = σ𝜅𝐧 ∙ 𝐧 = σ𝜅 (9)

Tangential stress balance

Taking dot product of Eq. (8) with any unit tangential vector t yields the tangential stress balance,

𝐧 ∙ (𝐓_𝟏 − 𝐓_𝟐) ∙ 𝐭 = ∇σ ∙ 𝐭 (10a)

The surface tension σ depends on temperature and composition of the interface, which can be treated as a constant. The gradient of surface tension will vanish and the tangential stress is continuous across the interface.

𝐧 ∙ (𝐓_𝟏 − 𝐓_𝟐) ∙ 𝐭 = 0 (10b)

Examples of Boundary conditions

Numerical approximation of the jump conditions The viscous stress tensor 𝛕 can be written as,

𝛕 = 𝜇 ∇𝐮 + ∇𝐮 ^𝑇 = 𝜇

𝛁𝑢

∇𝑣

∇𝑤

+ 𝜇

∇𝑢

∇𝑣

∇𝑤

𝑇

(11)

Using the jump notation 𝑥 = 𝑥₁ − 𝑥₂, and 𝐭_𝐼 and 𝐭_𝐼𝐼 the orthogonal unit tangential vectors, the stress jump conditions Eqs. (9) and (10b) can be rewritten as three separate jump conditions,

[𝑝 − 2𝜇(∇𝑢 ∙ 𝐧, ∇𝑣 ∙ 𝐧, ∇𝑤 ∙ 𝐧) ∙ 𝐧] = 𝜎𝜅 (12) 𝜇 ∇𝑢 ∙ 𝐧, ∇𝑣 ∙ 𝐧, ∇𝑤 ∙ 𝐧 ∙ 𝐭_𝐼 + 𝜇 ∇𝑢 ∙ 𝐭_𝐼, ∇𝑣 ∙ 𝐭_𝐼, ∇𝑤 ∙ 𝐭_𝐼 ∙ 𝐧 = 0 (13) 𝜇 ∇𝑢 ∙ 𝐧, ∇𝑣 ∙ 𝐧, ∇𝑤 ∙ 𝐧 ∙ 𝐭_𝐼𝐼 + 𝜇 ∇𝑢 ∙ 𝐭_𝐼𝐼, ∇𝑣 ∙ 𝐭_𝐼𝐼, ∇𝑤 ∙ 𝐭_𝐼𝐼 ∙ 𝐧 = 0 (14) The velocity is continuous and the tangential velocity derivatives are also continuous,

[∇𝐮 ∙ 𝐭_𝐼^𝑇] = 0 or ∇𝑢 ∙ 𝐭_𝐼 = ∇𝑣 ∙ 𝐭_𝐼 = ∇𝑤 ∙ 𝐭_𝐼 = 0 (15) [∇𝐮 ∙ 𝐭_𝐼𝐼^𝑇] = 0 or ∇𝑢 ∙ 𝐭_𝐼𝐼 = ∇𝑣 ∙ 𝐭_𝐼𝐼 = ∇𝑤 ∙ 𝐭_𝐼𝐼 = 0 (16)

18

Examples of Boundary conditions

Numerical approximation of the jump conditions

The normal stress condition can be written as,

[𝑝] − 2[𝜇](∇𝑢 ∙ 𝐧, ∇𝑣 ∙ 𝐧, ∇𝑤 ∙ 𝐧) ∙ 𝐧 = 𝜎𝜅 (17) The tangential jump conditions

[𝜇∇𝐮] = [𝜇](∇𝐮) 𝟎 𝐭_𝐼 𝐭_𝐼𝐼

𝑇 𝟎

𝐭_𝐼 𝐭_𝐼𝐼

+ 𝜇 𝐧^𝑻𝐧(∇𝐮)𝐧^𝑻𝐧 − [𝜇]

𝟎 𝐭_𝐼 𝐭_𝐼𝐼

𝑇 𝟎

𝐭_𝐼 𝐭_𝐼𝐼

∇𝐮 ^𝑇𝐧^𝑻𝐧 (18)

Note that the right-hand side of the above equation only involves velocity derivatives that are continuous across the interface. If 𝜇 = 0, then ∇𝐮 = 𝟎.

If the viscosity is smoothed to be continuous across the interface, the normal jump condition

[𝑝] = σ𝜅 (19)

The tangential viscous stress jump condition,

∇𝐮 ∙ 𝐧^𝑇 ∙ 𝐭_𝐼] + [ ∇𝐮 ∙ 𝐭_𝐼^𝑇 ∙ 𝐧 = 0 (20) According to Eq. (18), with a constant viscosity, all the velocity derivatives will be continuous across the interface which implies that both jump terms on the left hand side of the above equation are zero.

Examples of Boundary conditions

Vorticity condition across the interface

The vorticity in the normal direction is written as,

𝐧 ∙ 𝛚 = 𝐧 ∙ ∇ × 𝐮 = (𝐭_𝐼 × 𝐭_𝐼𝐼) ∙ ∇ × 𝐮 (21) Using the identity 𝐚 × 𝐛 ∙ 𝐜 × 𝐝 = 𝐚 ∙ 𝐜 𝐛 ∙ 𝐝 − 𝐚 ∙ 𝐝 𝐛 ∙ 𝐜 , Eq. (21) can be rewritten as,

𝐧 ∙ 𝛚 = 𝐭_𝐼 ∙ ∇ 𝐭_𝐼𝐼 ∙ 𝐮 − 𝐭_𝐼 ∙ 𝐮 𝐭_𝐼𝐼 ∙ ∇ (22) which is continuous across the interface since the right hand side of the above equation only involves tangential derivatives of the velocity.

The vorticity in the tangential directions,

𝐭_𝐼 ∙ 𝛚 = 𝐭_𝐼 ∙ ∇ × 𝐮 = 𝐭_𝐼𝐼 × 𝐧 ∙ ∇ × 𝐮 = 𝐭_𝐼𝐼 ∙ ∇ 𝐧 ∙ 𝐮 − 𝐭_𝐼𝐼 ∙ 𝐮 𝐧 ∙ ∇ (23) 𝐭_𝐼𝐼 ∙ 𝛚 = 𝐭_𝐼𝐼 ∙ ∇ × 𝐮 = 𝐧 × 𝐭_𝐼 ∙ ∇ × 𝐮 = 𝐧 ∙ ∇ 𝐭_𝐼 ∙ 𝐮 − 𝐧 ∙ 𝐮 𝐭_𝐼 ∙ ∇ (24) The tangential vorticities are generally not continuous across the interface since the normal derivatives, 𝐭_𝐼𝐼 ∙ 𝐮 𝐧 ∙ ∇ and 𝐧 ∙ ∇ 𝐭_𝐼 ∙ 𝐮 , are involved in the above equations, respectively.

However, as shown in Eq. (20), all the velocity derivatives will be continuous if the viscosity jump 𝜇 = 0, then tangential vorticities will also be continuous.

20

Examples of Boundary conditions

For two immiscible fluids with different density and viscosity:

Velocity, velocity gradient, viscosity, and shear stress distribution

Velocity, Uz, viscosity, and Taw profile for a layered two fluid flow.

Examples of Boundary conditions

Inlet/outlet/exit/outer/far-field BCs:

•

Inlet: V, p, T, specified, e.g., constant values are used, 𝑉 = 𝑉_𝑖𝑛, 𝑝 = 0, 𝑇 = 𝑇_𝑖𝑛,0

•

Outer or far-field: V, p, T, specified similarly as inlet

•

Exit: depends on the problems, but



often use U_xx = 0 and ^𝜕𝑝

𝜕𝑛 = 0.



For external flow, zero stream wise diffusion



For fully developed internal flow and wave problem, periodic BCs can be used



For unsteady internal flow, global mass conservation enforcement may be needed: 𝑈_𝑜𝑢𝑡 = 𝑈_𝑜𝑢𝑡 ^𝑄𝑖𝑛

𝑄_𝑜𝑢𝑡, where𝑄_𝑖𝑛 and 𝑄_𝑜𝑢𝑡 is the total inlet and outlet and flux, respectively.

Periodic Periodic

22

BCs in CFDShip-Iowa

BCs in CFDShip-Iowa

24

Simulation Examples using CFDShip-Iowa

Wave breaking in bump flow simulation: 2.2 billion grid points

Movie Plunging wave breaking:

• Inlet:

𝑢 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑣 = 𝑤 = 0

• Exit:

𝜕𝑢

𝜕𝑛

=

^𝜕𝑣

𝜕𝑛

=

^𝜕𝑤

𝜕𝑛

= 0

• For pressure,

^𝜕𝑝

𝜕𝑛

= 0 for all the boundaries.

• Mass balance needed at

the outlet.

Simulation Examples using CFDShip-Iowa

Wedge flow simulation, 1 billion grid points

Movie Wedge flow:

• The 𝑗

_𝑚𝑎𝑥

boundary is split into two parts: inlet and exit.

• Inlet:

𝑢 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑣 = 𝑤 = 0

• Exit:

𝜕𝑢

𝜕𝑛

=

^𝜕𝑣

𝜕𝑛

=

^𝜕𝑤

𝜕𝑛

= 0

• Slip BCs at both top and bottom.

• For pressure,

^𝜕𝑝

𝜕𝑛

= 0 for all the boundaries.

• Mass balance needed at the outlet.

Inlet Exit

Symmetric

No-Slip

26

Simulation Examples using CFDShip-Iowa

Stokes wave breaking: 3.2-12 Billion Grid Points

Movie Stokes wave breaking

Slip wall BCs at top and bottom

Periodic BCs at inlet, exit, and two sides.

Simulation Examples using CFDShip-Iowa

NSWC15E Planing Hull

Movies: bottom view side view

• Water is moving, ship-fixed system

• Inlet (10):

𝑢 = 𝑢

_{𝑖𝑛𝑓𝑙𝑜𝑤}

, 𝑣 = 𝑤 = 0

• Exit(11):

^𝜕2𝑈

𝜕𝑛²

= 0,

^𝜕𝑝

𝜕𝑛

= 0

28

Simulation Examples using CFDShip-Iowa

KCS free running

• Symmetric BC can not be used, use full ship

• Inlet (10): 𝑢 = 𝑣 = 𝑤 = 0,

since the ship is moving, earth fixed system (inertial)

• Exit (11)

Movies: free running